Subgroups — #1664

Order 128.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	1	yes	maximal, characteristic
2	#501	C₂	1	1	yes	maximal, characteristic, complemented
3	#491	C₂	1	1	yes	maximal, characteristic, complemented
4	#489	C₂	1	1	yes	maximal, characteristic
5	#330	C₂	1	1	yes	maximal, characteristic, complemented
6	#330	C₂	1	1	yes	maximal, characteristic, complemented
7	#329	C₂	1	1	yes	maximal, characteristic
8	C₄⋊Q₈	—	2	1	no	—
9	C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#126	—	2	1	no	—
11	#126	—	2	1	no	—
12	#125	C₂²	1	1	yes	characteristic
13	#125	C₂²	1	1	yes	characteristic
14	#124	C₂²	1	1	yes	characteristic
15	#123	—	2	1	no	—
16	C₂²⋊Q₈	—	2	1	no	—
17	C₂²⋊Q₈	—	2	1	no	—
18	#107	—	2	1	no	—
19	#106	—	2	1	no	—
20	#103	—	2	1	no	—
21	#103	—	2	1	no	—
22	#103	—	2	1	no	—
23	#102	—	2	1	no	—
24	#97	C₂²	1	1	yes	characteristic
25	#97	C₂²	1	1	yes	characteristic
26	#97	C₂²	1	1	yes	characteristic
27	C₂×Q₈	—	2	1	no	—
28	C₂×Q₈	—	2	1	no	—
29	C₂×Q₈	D₄	1	1	yes	characteristic
30	C₂×D₄	D₄	1	1	yes	characteristic
31	C₂²×C₄	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂²⋊C₄	—	2	1	no	—
48	C₂²⋊C₄	—	2	1	no	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	2	1	no	—
51	C₂²⋊C₄	—	2	1	no	—
52	C₄²	—	2	1	no	—
53	C₄²	C₂³	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	Q₈	—	4	1	no	—
56	Q₈	—	4	2	no	—
57	Q₈	—	8	1	no	—
58	D₄	—	4	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	8	1	no	—
66	C₂×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	4	1	no	—
71	C₈	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	#120	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	4	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	8	1	no	—
85	C₂	—	8	1	no	—
86	C₂	#454	1	1	yes	characteristic
87	C₂	#455	1	1	yes	characteristic
88	C₂	#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#120
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	C₂×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #501	C₂	semidirect
3. #491	C₂	semidirect
5. #330	C₂	semidirect
6. #330	C₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#563	2
#501	2
#491	2
#489	2
#330	2
#329	2